A note on the logistic equation subject to uncertainties in parameters
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A note on the logistic equation subject to uncertainties in parameters Fabio A. Dorini1 · Nara Bobko1 · Leyza B. Dorini2
Received: 6 September 2016 / Revised: 5 December 2016 / Accepted: 8 December 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract This paper discusses the logistic equation subject to uncertainties in the intrinsic growth rate, α, in the initial population density, N0 , and in the environmental carrying capacity, K . These parameters are treated as independent random variables. The random variable transformation method is applied to compute the first probability density function of the time–population density, N (t), and of its inflection point, t ∗ . Results for the density functions of N (t), for a fixed t > 0, and t ∗ are also provided for α, N0 and K uniformly distributed. Finally, numerical experiments illustrate the proposed theoretical results. Keywords Logistic equation · Uncertainties · Random variable transformation technique · First probability density function Mathematics Subject Classification Primary 34F05; Secondary 60H10
1 Introduction Over the last few years there has been an increased emphasis focused on differentiating and characterizing uncertainties that are inherently introduced in several mathematical models and simulation tools. Different stochastic population models were introduced to investigate
Communicated by Jose Alberto Cuminato.
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Fabio A. Dorini [email protected] Nara Bobko [email protected] Leyza B. Dorini [email protected]
1
Department of Mathematics, Federal University of Technology-Paraná, Curitiba, PR 80230-901, Brazil
2
Department of Informatics, Federal University of Technology-Paraná, Curitiba, PR 80230-901, Brazil
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the effect of environmental variability and perturbation (see Casabán et al. 2015, 2016; Holland et al. 2009; Kegan and West 2005; Lek 2007; Nasell 2003, for instance). In this paper, we explore uncertainties present in the logistic model, which finds applications in a wide range of fields such as sociology (diffusion of innovations), medicine (tumor growth rates), chemistry (reaction models) and ecology (population growth). This model was initially developed to describe population growth considering a self-limitation term that corrects the unlimited growth of the Malthusian model (Kot 2001). The classical logistic (or Verhulst’s) equation is the nonlinear initial value problem (IVP) d N (t) = α N (t) dt
1−
N (t) , t > 0, K
N (0) = N0 ,
(1)
where N (t) denotes the population density at time t, α > 0 is the intrinsic growth rate, N0 > 0 is the population density at time t = 0, and K > 0 is the environmental carrying capacity. Although these parameters are commonly considered accurate, the uncertainty quantification related to their implicit variation or randomness is essential to design meaningful and realistic models. Several approaches use random variables to represent the parameters α, N0 and K (Casabán et al. 2015, 2016; Kegan and West 2005; Wake and Wat
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